# The following questions are about the function f(x,y)=x^{2}e^{2xy}.

Kyran Hudson 2021-10-23 Answered
The following questions are about the function $f\left(x,y\right)={x}^{2}{e}^{2xy}$.
Find an equation of the tangent plane of f at the point (2,0, f(2,0))=(2,0,4).
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Step 1
If f(x,y) is a function of two variables x and y, then
${f}_{x}\left(x,y\right)=\frac{\partial f}{\partial x}$
${f}_{y}\left(x,y\right)=\frac{\partial f}{\partial y}$
The equation of the tangent plane of function f at the point $\left({x}_{0},{y}_{0}\right)$ is
$z=f\left({x}_{0},{y}_{0}\right)+{f}_{x}\left({x}_{0},{y}_{0}\right)\left(x-{x}_{0}\right)+{f}_{y}\left({x}_{0},{y}_{0}\right)\left(y-{y}_{0}\right)$
Step 2
The value of the given function at (2,0) is
$f\left(2,0\right)={\left(2\right)}^{2}{e}^{2\left(2\right)\left(0\right)}$
$=4{e}^{0}$
=4(1)
=4
The equation of the tangent plane of function f at the point (2,0) is
$z=f\left(2,0\right)+{f}_{x}\left(2,0\right)\left(x-2\right)+{f}_{y}\left(2,0\right)\left(y-0\right)$
z=4+4(x-2)+16(y)
z=4+4x-8+16y
z=4x+16y-4
Jeffrey Jordon